Q. A closed organ pipe of length L and open organ pipe contain gases of densities $p_{1}and\ p_{2}$ respectively. The compressibility of gases are equal in both the pipes. Both the pipes are vibrating in their first overtone with same frequency. The length of the open organ pipe is
A) $\frac{L}{3}$
B) $\frac{4L}{3}$
C) $\frac{4L}{3}\sqrt{\frac{p_{1}}{p_{2}}}$
D) $\frac{4l}{3}\sqrt{\frac{p_{2}}{p_{2}}}$
Explanation
(c) Given $v_{c} = v_{o}$(both first overtone)
Or
$3\left( \frac{v_{c}}{4L} \right) = 2\left( \frac{v_{o}}{2l_{o}} \right)$
$\therefore l_{o} = \frac{4}{3}\left( \frac{v_{o}}{v_{c}} \right)L = \frac{4}{3}\sqrt{\frac{\rho_{1}}{\rho_{2}}}L\ \ \ \ \ $
$\left( as\ v \propto \frac{1}{\sqrt{\rho}} \right)\ $
Therefore correct option is (c).
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.